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GeoGebra is open source software that combines dynamic geometry software, computer algebra system and spreadsheet functionality. This paper explores the current literature on each of these topics in an attempt to determine the advantages and barriers to using GeoGebra to enhance instruction in a secondary mathematics classroom.
Citation preview
GeoGebra in the Secondary Mathematics
Classroom 1
GeoGebra in the Secondary Mathematics Classroom:
A Literature Review
Dan Schellenberg
February, 2009
GeoGebra in the Secondary Mathematics
Classroom 2
GeoGebra in the Secondary Mathematics Classroom
GeoGebra is open source software that combines dynamic geometry software,
computer algebra system and spreadsheet functionality. This paper explores the current
literature on each of these topics in an attempt to determine the advantages and barriers to
using GeoGebra to enhance instruction in a secondary mathematics classroom.
The paper begins with an explanation of why technology is seen as an important part
of mathematics teaching and learning. This is followed by a description of the types of
technologies often used in mathematics teaching and learning. Specific attention is given to
dynamic geometry software, computer algebra systems and spreadsheets. Opportunities and
precautions are identified for each category of software. Finally, the important characteristics
of GeoGebra itself are examined.
Technology in Mathematics Education – Why Bother?
Technology plays an important part in the learning of mathematics. Students must
become familiar with the technological tools utilized in mathematics, whether that be an
abacus or a graphing calculator. Modern technology allows for easier exploration of
mathematics than was previously possible. “The speed of computers and calculators enables
students to produce many examples when exploring mathematical problems. This supports
the observation of patterns, and the making and justification of generalizations” (British
Education Communication Technology Agency, 2004, p. 1).
According to the National Council of Mathematics Teachers position statement
regarding technology, appropriate use of technology allows more students access to
mathematical concepts (National Council of Teachers of Mathematics, 2008). A motivating
factor for increasing the accessibility of mathematics is that mathematics knowledge has
become as an important part of critical citizenship (Adler, Ball, Krainer, Lin, & Novotna,
2005, p. 360). To help students gain the skills that will be useful as citizens, students must
GeoGebra in the Secondary Mathematics
Classroom 3
have the opportunity to use the same technology that is available outside the walls of their
classrooms. (Haapasalo, 2007, p. 9). Using the same technology that is available outside the
classroom allows students to transfer their knowledge into the world as they move beyond
formal education.
Some teachers and school systems remain wary of integrating technology into
mathematics education. The three most common reasons are curriculum scope (convincing
teachers the benefits are worth the change), availability of the technology (open computer
labs, for example) and accessibility of the programs (technology that is easy enough to learn
that the focus remains on the math) (Little, 2008, p. 49). Equipment failure can also be a
major roadblock to the adoption of technology, as teachers will not commit to using
something they cannot rely on in their daily teaching (Cuban, Kirkpatrick, & Peck, 2001, p.
829).
The views of the mathematics teacher greatly influence whether and how technology
will be incorporated into the classroom. According to a recent study, middle-aged and more
experienced teachers were more likely to integrate technology than their younger
counterparts, despite having a more negative attitude regarding technology (Hung & Hsu,
2007, p. 233). This suggests that familiarity with technology might not correlate to increased
technology use in the classroom. A base level of technical skill is required, however, as a
previous study notes that “effective teachers who use ICT [information and communications
technology] are teachers who are confident with ICT” (Bramald, Miller, & Higgins, 2000, p.
5).
Types of Technology Used in Mathematics Education
The technology used in mathematics teaching and learning can be categorized into
two major types, virtual manipulatives and general software tools (Preiner, 2008, p. 26). A
virtual manipulative can be defined as “an interactive, Web-based visual representation of a
GeoGebra in the Secondary Mathematics
Classroom 4
dynamic object that presents opportunities for constructing mathematical knowledge”
(Moyer, Bolyard, & Spikell, 2002). Virtual manipulatives allow a student to interact with the
mathematical situation without any additional skills or training required, though the student’s
exploration is limited by the design of the virtual manipulative. By contrast, general software
tools allow the student to explore any number of mathematical concepts, but require some
training to use.
A variety of general software tools are used in mathematics, including dynamic
geometry software, computer algebra systems and spreadsheets. Barzel
defines general software tools as “tools [that] can be used for a wide set of tasks and
be considered to be general purpose tools that are not useful for only a limited number of
specific tasks – that is the character and as well the most important benefit of general tools”
(2007, p. 81).
The remainder of this literature review will be spent on examining the research on
dynamic geometry software, computer algebra software and spreadsheets. These are the
types of software that GeoGebra seeks to integrate into one coherent tool.
Dynamic Geometry Software
Dynamic Geometry Software (DGS) is the most easily adopted form of general
software tools, as it was explicitly designed for classroom use (Ruthven, 2008, p. 1). DGS is
controlled primarily with the mouse, allowing the basic functionality to be easily learned.
Using DGS, teachers and students are able to quickly and accurately explore geometrical
figures, changing their dimensions while maintaining the mathematical relationships in the
figure. For example, a figure could be drawn showing a perpendicular bisector of a line
segment. As the line segment is dragged, changing its position and length, the perpendicular
bisector automatically moves as well. The important features of DGS are listed by Kokol-
Voljic as:
GeoGebra in the Secondary Mathematics
Classroom 5
- a dynamic modeling of the traditional paper and pencil (blackboard and chalk)
teaching environment through the drag mode
- an option to condense a sequence of commands to form a "new command", a
macro
- an option to visualize the paths of the movements of geometrical objects, a locus
(2007, p. 56)
Figure 1. Screenshot of dynamic geometry software Geometer’s Sketchpad.
DGS has the ability to profoundly change the way we teach proof, one of the most
crucial ideas in mathematics. DGS allows students to instantly create and test their
conjectures, allowing them the freedom to explore geometry and discover patterns. Although
students can easily find patterns using DGS, researchers are advising users of DGS to use
exploration merely as the foundation for deductive proof, since some teachers have begun to
use exploration as a replacement for proof (Hanna, 2000, p. 14). Teachers’ tendency to
replace formal proof with dynamic exploration is seen as a reaction to improper use of formal
proof, such as only proving things students are already convinced is true (Hoyles & Jones,
1998, p. 122).
GeoGebra in the Secondary Mathematics
Classroom 6
Although the power and flexibility of DGS is enticing, we must understand and
acknowledge that changing the medium of teaching geometry will cause important changes in
the way students construct meaning about geometry. Jones lists a number of specific areas in
which DGS has a mediational impact, including:
- The students’ understanding that the order in which objects were created leads to a
hierarchy of functional dependency within a figure.
- The constraint of robustness of a figure under drag becoming linked with using
points of intersection to try to hold the figure together.
- The ‘dynamic’ nature of the software influencing the form of explanation given by
the students. (Jones, 2000, p. 80)
The role of the teacher is shifted when DGS is utilized in the classroom, but the
teacher’s role remains critically important; the teacher’s guidance is crucial as the student
tries to construct meaning from the explorations they are involved in.
The artefact [DGS] is exploited by a double use, with respect to which it functions as
semiotic mediator. On the one hand, meanings emerge from the activity – the learner
uses the artefact in actions aimed at accomplishing a certain task; on the other hand,
the teacher uses the artefact to direct the development of meanings that are
mathematically consistent. (Mariotti, 2000, p. 37)
There are pitfalls inherent with free exploration in DGS, such as students
inadvertently creating a special case by dragging a generic drawing (Sinclair, 2003, p. 291).
This could lead students to incorrect assumptions about mathematical figures, such as
thinking that the Pythagorean theorem holds for all triangles, when in fact it is only true with
right triangles. While these pitfalls should not stop us from using DGS, we must be aware of
them as we begin to incorporate the use of DGS in our classrooms.
GeoGebra in the Secondary Mathematics
Classroom 7
Computer Algebra System
Another type of general software being used in mathematics education is a Computer
Algebra System (CAS). A CAS can be defined as “a piece of software which is capable of
working symbolically as well as numerically. In principle it is a program which does on a
computer the manipulation that has traditionally been done with pencil and paper” (Lawson,
1997, p. 228). CAS are primarily controlled by the keyboard through textual and numerical
input. It is important to note that CAS was created for use by practicing mathematicians, not
for mathematics education (Ruthven, 2008, p. 1). This has caused slower adoption of CAS
into the classroom, and teachers and researchers are still attempting to come to terms with the
effects of using CAS in the classroom. Much of the discussion on CAS in the classroom
revolves around what portions of the curriculum students need to know how to do by hand,
and what portions they can off-load to a computer. The answers to these questions greatly
influence what is taught, and how it is assessed.
Figure 2. Screenshot of factoring with the computer algebra system Mathematica.
Supporters of CAS in education emphasize the ability of students to access higher
level concepts, without having to drudge through tedious algebraic manipulations (Atiyah,
Monaghan, & Pierce, 2004, p. 157). Access to these higher-level concepts allows students to
leave contrived problems behind, giving them a chance to explore real world situations
GeoGebra in the Secondary Mathematics
Classroom 8
instead (Heid & M. T. Edwards, 2001, p. 128). Leigh-Lancaster as gives a fairly
comprehensive list of possible benefits resulting from the use of CAS, including
- the possibility for improved teaching of traditional mathematical topics
- opportunities for new selection and organization of mathematical topics
- access to important mathematical ideas that have previously been to difficult to
teach effectively
- a vehicle for mathematical discovery
- long and complex calculations can be carried out by the technology, enabling
students to concentrate on the conceptual aspects of mathematics
- the technology provides immediate feedback so that students can independently
monitor and verify their ideas
- the need to express mathematical ideas in a form understood by the technology
helps students to clarify their mathematical thinking
- situations and problems can be modeled in more complex and realistic ways
(2003, p. 5)
Despite the perceived benefits of CAS, some researchers find fault with the
underlying assumption that concepts and skills can be separated. While this does not
necessarily lead them to reject the notion of using CAS in the classroom, it does change the
way in which CAS is used. The French researcher Lagrange is among the leading voices in
this camp. For Lagrange, manual skills (or more generally, techniques) are required for the
student to construct meaning. In my own experiences as a classroom teacher, I have found
that if students are shown a technological solution before having a chance to practice a
technique by hand, they may not ever truly understand the nature of what the technology is
doing. For example, if a student is taught to multiply matrices using a graphing calculator,
they may be very proficient at typing the numbers into the calculator, but may have no idea
GeoGebra in the Secondary Mathematics
Classroom 9
about how to interpret the elements of the resulting matrix. I have found it to be much more
effective to introduce matrix multiplication by guiding the students through a word problem
and having the students define matrix multiplication themselves. In the words of Lagrange,
At certain moments a technique can take the form of a skill. This is particularly
the case when a certain ‘routinisation’ is necessary… It is certain that the availability
of new instruments reduces the urgency of this routinisation… But techniques must
not be considered only in their routinised form. The work of constituting techniques in
response to tasks, and of theoretical elaboration on the problems posed by these
techniques remains fundamental to learning. (Lagrange, as cited in Ruthven, 2002, p.
16)
A more pragmatic concern with the use of CAS in the classroom is that students may
be confused by the results given by the CAS (Artigue, 2002, p. 265). For example, when a
secondary mathematics student is taught to factor a difference of cubes, they are taught a
rigid algorithm, which will result in all students achieving the same answer. The CAS may or
may not represent the factored form of the expression in the same manner the student is used
to seeing. A student working by hand would factor as follows:
€
8x 3 − 27( )
= 2x − 3( ) 4x 2 +12x + 9( )
When the CAS feature of GeoGebra performs this factorization, the result is:
€
8x 3 − 27( )
= 12x + 8x 2 +18( ) x − 3/2( )
Although these expressions are in fact equivalent, recognizing that fact may not be trivial for
a student without the ability to perform such tasks mentally or by hand. This sort of situation
can lead to students being unable to determine if the answers given by the CAS are
reasonable (Waits & Demana, 1998).
GeoGebra in the Secondary Mathematics
Classroom 10
The power of CAS will fundamentally change mathematics learning and assessment.
“Whereas graphics calculators, for many teachers, slotted easily into the curriculum and
enhanced their teaching with little threat, CAS demands a more thorough response” (Kendal,
Stacey, & Pierce, 2005, p. 105). Paper and pencil algorithms (techniques, in the terminology
of Lagrange) must be examined individually to determine if they contribute understanding for
the student. Algorithms that do not contribute to a student’s understanding should be
performed with technology (Waits & Demana, 1998).
Spreadsheets
Another type of general mathematics software is the spreadsheet. A spreadsheet is
simply an array of rows and columns that allow calculations to be quickly performed. More
recently, “the basic paradigm of an array of rows-and-columns with automatic update and
display of results has been extended with libraries of mathematical and statistical functions
[and] versatile graphing and charting facilities” (Baker & Sugden, 2003, p. 19). This
extension of the functionality of spreadsheets has allowed spreadsheets to become useful
when teaching a variety of mathematical topics.
Figure 3. Screenshot of spreadsheet software Microsoft Excel.
Spreadsheets are useful tools for exploring a large range of mathematical topics. One
of the simplest uses of the spreadsheet at a secondary level occurs when teaching statistics.
GeoGebra in the Secondary Mathematics
Classroom 11
Although spreadsheets may not be suited to deal with in-depth mathematical statistics, they
can be very useful for introductory level statistics, as would be seen in a secondary
mathematics curriculum (Nash, 2008, p. 4603). Performing simple calculations on statistical
data becomes a trivial with a spreadsheet. Spreadsheets can also be used in teaching such
diverse mathematical topics as inequalities (Abramovich, 2005), limits in calculus
(Abramovich & Levin, 1994) and the concept of infinity (Abramovich & Norton, 2000).
Studies have shown increased student understanding of statistical graphs as a result of
using spreadsheet explorations in statistics (Wu & YoongWong, 2007). As the use of
spreadsheets in the classroom becomes more complex, however, new issues arise. Unless
they are taught otherwise, students tend to create spreadsheets that are not reliable when cell
values are changed. Students must be taught how to create spreadsheets that can solve
general problems, instead of only being useful for only one specific case (Niess, 2006, p.
199).
GeoGebra’s Defining Features
GeoGebra is software that attempts to combine DGS, CAS and spreadsheets into one
application. “On the one hand, GeoGebra is a dynamic geometry system in which you work
with points, vectors, segments, lines, and conic sections. On the other hand, equations and
coordinates can be entered directly” (Sangwin, 2007, p. 36). Every object in GeoGebra has a
representation in both the algebra window, as well as the geometry window. The user can
adjust the value of the object through either representation, allowing them to either drag the
geometric figure using the mouse, or change the symbolic representation using the keyboard.
While GeoGebra attempts to combine aspects of DGS, CAS and spreadsheets, small
annoyances reveal that this combination is done imperfectly. One such annoyance is the need
to learn arcane syntax in order to show dynamic text or calculations when using the DGS
feature of the software. For example, to create dynamic text that updates as the location of a
GeoGebra in the Secondary Mathematics
Classroom 12
point changes, one would have to enter something like ”a = ” + a + ”cm”. By
contrast, the ease of use of Geometer’s Sketchpad (another popular DGS) when doing the
same task allows students to show dynamic calculations without having to learn any syntax at
all. Some of these annoyances may be a result of GeoGebra still being relatively new
software, having been initially created by Markus Hohenwarter in 2001 as part of his
Master’s thesis in mathematics education (Preiner, 2008, p. 36).
Figure 4. Screenshot of finding the area of a triangle with GeoGebra.
The CAS abilities of GeoGebra are currently quite limited, though in the pre-release
version of GeoGebra the CAS aspects of the software have been dramatically enhanced. In a
recent post to the GeoGebra CAS mailing list, Markus Hohenwarter explained that
development version (pre-release version) of GeoGebra now includes a full featured CAS
system by incorporating an open source CAS into GeoGebra (M. Hohenwarter, 2008).
GeoGebra in the Secondary Mathematics
Classroom 13
The development version of GeoGebra also incorporates spreadsheet functionality,
though certain limitations exist. Currently, only 100 rows of data can be viewed in the
spreadsheet mode. Spreadsheet cell ranges must be typed when performing calculations, not
selected with the mouse. For example, one can type Mean[A1:A9], but you cannot type
Mean[] and then highlight which cells you want to calculate the median of. Many advanced
features of popular spreadsheet applications such as Excel are not available in GeoGebra,
though most functions that would be used at a high school level are already available.
GeoGebra is an open source application, which gives GeoGebra both moral and
pragmatic benefits over proprietary software. GeoGebra is freely available to schools and
students, eliminating the cost factor for schools with limited budgets. Students are able to use
the application at home on their private computers with no site licensing concerns (M.
Hohenwarter, J. Hohenwarter, Kreis, & Lavicza, 2008, p. 2). Markus Hohenwarter, the
creator of GeoGebra, has stated the reason GeoGebra is released as a free, open source
application is that he believes education should be free (Edwards & Jones, 2006). A side
benefit of being open source is that a development community has grown around the project,
which has allowed GeoGebra to be translated into many languages (39 different languages as
of GeoGebra 3.0 in March 2008), making it accessible to many more students and educators.
GeoGebra is written in Java, which allows it to run on virtually any platform (Mac,
Windows, Linux). Being written in Java also allows GeoGebra to easily export files as
dynamic webpages. This allows for simple creation of online math explorations, often called
math applets. The ease with which GeoGebra sketches can be shared online has lead to a
community of teachers using GeoGebra freely sharing their resources with one another online
at http://www.geogebra.org/en/wiki.
While an online wiki of resources is useful for early adopters of GeoGebra, the
majority of teachers will not begin using GeoGebra on the basis of resources being available
GeoGebra in the Secondary Mathematics
Classroom 14
online. Simply providing technology to teachers does not lead to successful integration of
that technology in their teaching (Cuban et al., 2001). To address this concern, an
International GeoGebra Institute has been created, with the purpose of providing structured
training and support to teachers interested in using GeoGebra (M. Hohenwarter & Lavicza,
2007). Various chapters have been set up across the world, allowing those interested in
learning GeoGebra to have local support.
Conclusion
Incorporating technology into mathematics teaching and learning allows greater
access to mathematical concepts. General mathematics software allows students to explore
any number of mathematical situations, but require students to learn the software first.
Dynamic Geometry Software is quite easy to use, allowing students and teachers to test
conjectures by exploring geometrical figures. The manner in which proof is taught in
mathematics has been greatly affected by the introduction of DGS to the classroom.
Computer Algebra Systems are able to perform much of the symbolic manipulation that
students do by hand. Educators must determine which algorithms can be delegated to a CAS
and which must be done by hand. Spreadsheets are particularly useful when teaching
statistics, but can also be used to teach a wider variety of mathematical topics.
GeoGebra combines DGS and CAS into one application (development/future versions
also include a spreadsheet). GeoGebra is open source software, which allows anyone to
download the software and use it for free. As a Java application, it can run on any platform.
To help teachers learn how to incorporate GeoGebra into their classrooms, an International
GeoGebra Institute has been created to provide structured training and support.
GeoGebra in the Secondary Mathematics
Classroom 15
References
Abramovich, S. (2005). Inequalities and spreadsheet modeling. Spreadsheets in Education,
2(1), 1–21.
Abramovich, S., & Levin, I. (1994). Spreadsheets in teaching and learning topics in calculus.
International Journal of Mathematical Education in Science and Technology, 25(2),
263. doi: 10.1080/0020739940250213.
Abramovich, S., & Norton, A. (2000). Technology-enabled pedagogy as an informal link
between finite and infinite concepts in secondary mathematics. The Mathematics
Educator, 10(2), 31-46.
Adler, J., Ball, D., Krainer, K., Lin, F. L., & Novotna, J. (2005). Reflections on an emerging
field: researching mathematics teacher education. Educational Studies in
Mathematics, 60(3), 359-381. doi: 10.1007/s10649-005-5072-6.
Artigue, M. (2002). Learning mathematics in a CAS environment: the genesis of a reflection
about instrumentation and the dialectics between technical and conceptual work.
International Journal of Computers for Mathematical Learning, 7(3), 245-274. doi:
10.1023/A:1022103903080.
Atiyah, M., Monaghan, J., & Pierce, R. (2004). Computer algebra systems and algebra:
curriculum, assessment, teaching, and learning. In The future of the teaching and
learning of algebra: The 12th ICMI study (pp. 153-186). Boston: Kluwer Academic
Publishers. Retrieved January 12, 2009, from http://dx.doi.org/10.1007/1-4020-8131-
6_7.
Baker, J. E., & Sugden, S. J. (2003). Spreadsheets in Education–The First 25 Years.
Spreadsheets in Education, 1(1), 18-43.
Barzel, B. (2007). New technology? New ways of teaching - no time left for that!
International Journal for Technology in Mathematics Education, 14(2), 77-90.
GeoGebra in the Secondary Mathematics
Classroom 16
Bramald, R., Miller, J., & Higgins, S. (2000). ICT, mathematics and effective teaching.
Mathematics Education Review, 12, 1–13.
British Education Communication Technology Agency. (2004). Using web-based resources
in secondary mathematics. British Education Communication Technology Agency.
Retrieved January 10, 2009, from http://foi.becta.org.uk/display.cfm?
cfid=1476190&cftoken=29154&resID=36065.
Cuban, L., Kirkpatrick, H., & Peck, C. (2001). High access and low use of technologies in
high school classrooms: Explaining an apparent paradox. American Educational
Research Journal, 38(4), 813-834.
Edwards, J. A., & Jones, K. (2006). Linking geometry and algebra with GeoGebra.
Mathematics Teaching, 194, 28-30.
Haapasalo, L. (2007). Adapting mathematics education to the needs of ICT. The Electronic
Journal of Mathematics and Technology, 1(1), 1-10.
Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in
Mathematics, 44(1), 5-23. doi: 10.1023/A:1012737223465.
Heid, M. K., & Edwards, M. T. (2001). Computer algebra systems: revolution or retrofit for
today's mathematics classrooms? Theory Into Practice, 40(2), 128 - 136. doi:
10.1207/s15430421tip4002_7 .
Hohenwarter, M. (2008, December 6). Ted Kosan and MathPiper. geogebra-cas. Retrieved
February 19, 2009, from http://www.freelists.org/post/geogebra-cas/Ted-Kosan-and-
MathPiper.
Hohenwarter, M., Hohenwarter, J., Kreis, Y., & Lavicza, Z. (2008). Teaching and learning
calculus with free dynamic mathematics software GeoGebra. Monterrey, Nuevo Leon,
Mexico.
GeoGebra in the Secondary Mathematics
Classroom 17
Hohenwarter, M., & Lavicza, Z. (2007). Mathematics teacher development with ICT:
Towards an international GeoGebra Institute. Proceedings of the British Society for
Research into Learning Mathematics, 27(3). Retrieved December 20, 2008, from
http://www.bsrlm.org.uk/IPs/ip27-3/BSRLM-IP-27-3-09.pdf.
Hoyles, C., & Jones, K. (1998). Proof in dynamic geometry contexts. In C. Mammana & V.
Villani (Eds.), Perspectives on the teaching of geometry for the 21st century (pp. 121-
128). Dordrecht: Kluwer. Retrieved January 11, 2009, from
http://eprints.soton.ac.uk/41227/.
Hung, Y., & Hsu, Y. (2007). Examining teachers' CBT use in the classroom: A study in
secondary schools in Taiwan. Journal of Educational Technology & Society, 10(3),
233-246.
Jones, K. (2000). Providing a foundation for deductive reasoning: students' interpretations
when using dynamic geometry software and their evolving mathematical
explanations. Educational Studies in Mathematics, 44(1-2), 55-85. doi:
10.1023/A:1012789201736.
Kendal, M., Stacey, K., & Pierce, R. (2005). The influence of a computer algebra
environment on teachers’ practice. In The didactical challenge of symbolic
calculators (pp. 83-112). Retrieved January 24, 2009, from
http://dx.doi.org/10.1007/0-387-23435-7_5.
Kokol-Voljc, V. (2007). Use of mathematical software in pre-service teacher training: The
case of DGS. Proceedings of the British Society for Research into Learning
Mathematics, 27(3), 55.
Lagrange, J. (2000). L'intégration d'instruments informatiques dans l'enseignement: Une
approche par les techniques. Educational Studies in Mathematics, 43(1), 1-30. doi:
10.1023/A:1012086721534.
GeoGebra in the Secondary Mathematics
Classroom 18
Lawson, D. (1997). The challenge of computer algebra to engineering mathematics.
Engineering Science and Education Journal, 6(6), 228-232.
Leigh-Lancaster, D. (2003). The Victorian curriculum and assessment authority
mathematical methods computer algebra pilot study and examinations (p. 34).
Reports - Research, Rheims, France. Retrieved January 12, 2009, from
http://www.eric.ed.gov/ERICWebPortal/contentdelivery/servlet/ERICServlet?
accno=ED480463.
Little, C. (2008). Interactive geometry in the classroom: old barriers and new opportunities.
In M. Joubert (Ed.), Proceedings of the British Society for Research into Learning
Mathematics, 2 (Vol. 28, pp. 49-54). University of Southampton.
Mariotti, M. (2000). Introduction to proof: The mediation of a dynamic software
environment. Educational Studies in Mathematics, 44(1), 25-53. doi:
10.1023/A:1012733122556.
Moyer, P., Bolyard, J., & Spikell, M. (2002). What are virtual manipulatives? Teaching
Children Mathematics, 8(6), 372.
Nash, J. C. (2008). Teaching statistics with Excel 2007 and other spreadsheets.
Computational Statistics & Data Analysis, 52(10), 4602-4606. doi:
10.1016/j.csda.2008.03.008.
National Council of Teachers of Mathematics. (2008, March). The role of technology in the
teaching and learning of mathematics: A position of the national council of teachers
of mathematics. Retrieved January 10, 2009, from
http://www.nctm.org/about/content.aspx?id=14233.
Niess, M. L. (2006). Guest editorial: preparing teachers to teach mathematics with
technology. Contemporary Issues in Technology and Teacher Education, 6(2), 195-
203.
GeoGebra in the Secondary Mathematics
Classroom 19
Preiner, J. (2008, April 2). Introducing dynamic mathematics software to mathematics
teachers: the case of GeoGebra. University of Salzburg. Retrieved December 9, 2008,
from http://www.geogebra.org/publications/jpreiner-dissertation.pdf.
Ruthven, K. (2002). Instrumenting mathematical activity: reflections on key studies of the
educational use of computer algebra systems. International Journal of Computers for
Mathematical Learning, 7(3), 275-291. doi: 10.1023/A:1022108003988.
Ruthven, K. (2008). The interpretative flexibility, instrumental evolution and
institutional adoption of mathematical software in educational practice: the
examples of computer algebra and dynamic geometry. New York. Retrieved January
24, 2009, from
http://www.educ.cam.ac.uk/events/conferences/annual/camera/camera2008/papers/
Ruthven_CamERA08_paper.pdf.
Sangwin, C. (2007). A brief review of GeoGebra: dynamic mathematics. MSOR Connections,
7(2), 36.
Sinclair, M. (2003). Some implications of the results of a case study for the design of pre-
constructed, dynamic geometry sketches and accompanying materials. Educational
Studies in Mathematics, 52(3), 289-317. doi: 10.1023/A:1024305603330.
Waits, B. K., & Demana, F. (1998). The role of hand-held computer symbolic algebra in
mathematics education in the twenty-first century:
A call for action! Retrieved January 13, 2009, from
http://mathforum.org/technology/papers/papers/waits/waits.html.
Wu, Y., & YoongWong, K. (2007). Impact of a spreadsheet exploration on secondary school
students' understanding of statistical graphs. The Journal of Computers in
Mathematics and Science Teaching, 26(4), 355-385.